ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  2omotap GIF version

Theorem 2omotap 7257
Description: If there is at most one tight apartness on 2o, excluded middle follows. Based on online discussions by Tom de Jong, Andrew W Swan, and Martin Escardo. (Contributed by Jim Kingdon, 6-Feb-2025.)
Assertion
Ref Expression
2omotap (∃*𝑟 𝑟 TAp 2oEXMID)

Proof of Theorem 2omotap
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 2omotaplemst 7256 . . . . 5 ((∃*𝑟 𝑟 TAp 2o ∧ ¬ ¬ 𝑥 = {∅}) → 𝑥 = {∅})
21ex 115 . . . 4 (∃*𝑟 𝑟 TAp 2o → (¬ ¬ 𝑥 = {∅} → 𝑥 = {∅}))
3 df-stab 831 . . . 4 (STAB 𝑥 = {∅} ↔ (¬ ¬ 𝑥 = {∅} → 𝑥 = {∅}))
42, 3sylibr 134 . . 3 (∃*𝑟 𝑟 TAp 2oSTAB 𝑥 = {∅})
54adantr 276 . 2 ((∃*𝑟 𝑟 TAp 2o𝑥 ⊆ {∅}) → STAB 𝑥 = {∅})
65exmid1stab 4208 1 (∃*𝑟 𝑟 TAp 2oEXMID)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  STAB wstab 830   = wceq 1353  ∃*wmo 2027  wss 3129  c0 3422  {csn 3592  EXMIDwem 4194  2oc2o 6410   TAp wtap 7247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-tr 4102  df-exmid 4195  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-1o 6416  df-2o 6417  df-pap 7246  df-tap 7248
This theorem is referenced by:  exmidmotap  7259
  Copyright terms: Public domain W3C validator